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The fundamental problem in the study of parallel-server systems is that of finding and analyzing routing policies of arriving jobs to the servers that efficiently balance the load on the servers. The most well-studied policies are (in decreasing order of efficiency) join the shortest workload (JSW), which assigns arrivals to the server with the least workload; join the shortest queue (JSQ), which assigns arrivals to the smallest queue; the power-of-[Formula: see text] (PW([Formula: see text])), which assigns arrivals to the shortest among [Formula: see text] queues that are sampled from the total of [Formula: see text] queues uniformly at random; and uniform routing, under which arrivals are routed to one of the [Formula: see text] queues uniformly at random. In this paper we study the stability problem of parallel-server systems, assuming that routing errors may occur, so that arrivals may be routed to the wrong queue (not the smallest among the relevant queues) with a positive probability. We treat this routing mechanism as a probabilistic routing policy, named a [Formula: see text]-allocation policy, that generalizes the PW([Formula: see text]) policy, and thus also the JSQ and uniform routing, where [Formula: see text] is an [Formula: see text]-dimensional vector whose components are the routing probabilities. Our goal is to study the (in)stability problem of the system under this routing mechanism, and under its “nonidling” version, which assigns new arrivals to an idle server, if such a server is available, and otherwise routes according to the [Formula: see text]-allocation rule. We characterize a sufficient condition for stability, and prove that the stability region, as a function of the system’s primitives and [Formula: see text], is in general smaller than the set [Formula: see text]. Our analyses build on representing the queue process as a continuous-time Markov chain in an ordered space of [Formula: see text]-dimensional real-valued vectors, and using a generalized form of the Schur-convex order.
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This content will become publicly available on August 29, 2025
Robust Online Selection with Uncertain Offer Acceptance
Online advertising has motivated interest in online selection problems. Displaying ads to the right users benefits both the platform (e.g., via pay-per-click) and the advertisers (by increasing their reach). In practice, not all users click on displayed ads, while the platform’s algorithm may miss the users most disposed to do so. This mismatch decreases the platform’s revenue and the advertiser’s chances to reach the right customers. With this motivation, we propose a secretary problem where a candidate may or may not accept an offer according to a known probability p. Because we do not know the top candidate willing to accept an offer, the goal is to maximize a robust objective defined as the minimum over integers k of the probability of choosing one of the top k candidates, given that one of these candidates will accept an offer. Using Markov decision process theory, we derive a linear program for this max-min objective whose solution encodes an optimal policy. The derivation may be of independent interest, as it is generalizable and can be used to obtain linear programs for many online selection models. We further relax this linear program into an infinite counterpart, which we use to provide bounds for the objective and closed-form policies. For [Formula: see text], an optimal policy is a simple threshold rule that observes the first [Formula: see text] fraction of candidates and subsequently makes offers to the best candidate observed so far. Funding: Financial support from the U.S. National Science Foundation [Grants CCF-2106444, CCF-1910423, and CMMI 1552479] is gratefully acknowledged.
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- PAR ID:
- 10548931
- Publisher / Repository:
- INFORMS
- Date Published:
- Journal Name:
- Mathematics of Operations Research
- ISSN:
- 0364-765X
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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